Theorem 1.2 ([1]}, [2]}) 

Let \(A \subset \mathbb {F}^n \times \mathbb {F}^n\)  be of density \(\alpha \)  and let \(w = \textup {h}\textup {h}\textup {v}\textup {v}\textup {h}\textup {h}\) . Then there exists a bilinear variety \(B \subset \phi _w(A)\) 
of codimension \(r=\exp (\exp (\log ^{O(1)}\alpha ^{-1}))\) .

